Real numbers and complex numbers

[FactChecker]

I like the idea of a principal square root of a positive real (that's the one I was assuming in my post for sqrt(2), and I have heard about that) better than the idea of a "principal" complex square root of a negative number. In fact the more I think about it, it seems to me there is really no way to define a principal square root of a negative number without referring to a specific model of the complex numbers, one in which a preferred square root of -1 has been chosen and given a specific name. I.e. in the abstract complex numbers, defined merely as an algebraic closure of the reals, there is no preferred square root of -1, hence no distinguished element to call i.

When I defined the principal root in terms of "counter clockwise" motion on the unit circle, I was assuming we are taking as our model of the complex numbers the usual model defined by the set of ordered pairs of reals, and in which one designates the pair (0,1) as i. But there is no intrinsic reason to choose this as i, rather than (0,-1). So if our definition of the complex numbers is only an algebraically closed field, algebraic over the reals, there is no distinguished element i.

That sounds right, but there is practically no area of mathematics where there are not canonical representations and standard forms that are somewhat arbitrary. What version of this you prefer probably depends on where you are going with it. In complex analysis, applied math, and engineering, there are good reasons to introduce the standard definitions of i and the principle branch of the square root, logarithm, etc.

 

[swampwiz]

Summary:: Why don't the answers agree?

To find √(-2)√(-3).

Method 1.
√(-2)√(-3) = √( (-2)(-3) ) = √(6).

Method 2.
√(-2)√(-3) = √( (-1)(2) )√( (-1)(3) ) = √((-1)√(2)√(-1)√(3) = i√(2)i√(3) = (i)(i)√(2)√(3) = -1√( (2)(3) ) =-√6.

Why don't the two methods give the same answer?
Thanks for any help.

There are a few mistakes. And there is also the ambiguity of what is exactly meant by the radical. Technically, a radical could indicate any deMoivre root (which for the square root means one is the ± of the other), but a common usage is for any radical to represent the value that has a complex angle of the product of complex angle of the base and the reciprocal of the root degree - aka the principle root.

If the radical is to be regarded as the former, then there is no problem since both results are the same (i.e., ± of the principal square root of 6), and the steps just have a extra ± factors that are all redundant.

If the radical is to be regarded as the latter, then the proper analysis, using phasor notation (i.e., the phasor has the value of the complex exponent of e to the product of i & the angle (as radians), would be:

[ √(-2)√(-3) ] = [ 2 (∠180°) ]^1/2 [ 3 (∠180°) ]^1/2 ]

= [ 2^1/2 (∠90°) ] [ 3^1/2 (∠90°) ] = [ 6^1/2 (∠180°) ]

= - 6^1/2

Thus this step in Method 1 is incorrect; you cannot bring everything under the radical like you could under a general exponent.

√(-2)√(-3) = √( (-2)(-3) ) -> INCORRECT

[ (-2)^a (-3)^a ] = { (-2)(-3) }^a -> OK, if a is an integer

Method 2 is loaded with the same type of mistake, although obviously in the end, the result is correct.

 

[Svein]

Actually, taking a (square) root of something in the complex plane is not straightforward at all. It involves analytic continuation and the concept of Riemann surfaces. A short example involving the square root of two:
$2=2e^{2n\pi}$ where n is any integer. Therefore $\sqrt{2}=\sqrt{2}e^{n\pi}$ where n is any integer. Since $e^{n\pi}=1$ for any even n and $e^{n\pi}=-1$ for any odd n, the square root is not a function in the classical sense.

Generalizing slightly, $\sqrt{z}$ is only single-valued for z=0. We can, however, define several single-valued functions for that square root by restricting arg(z) suitably. Wherever two of those functions are defined on a common interval for arg(z), they are identical and thus are analytic continuations of each other.

Granted, this is just a one-paragraph sketch of a subject requiring several pages to be anywhere near complete.

 

[wrobel]

Summary:: Why don't the answers agree?

To find √(-2)√(-3).

this statement seems to be incorrect by itself. $\sqrt{z}$ is a multivalued function
which branch is expected to choose?